On the $p$-adic variation of Heegner points

TL;DR

This paper establishes an explicit reciprocity law linking big Heegner points to a new two-variable p-adic L-function, with applications to Heegner cycles and Selmer groups of CM elliptic curves, advancing understanding of p-adic variation and BSD conjecture.

Contribution

It introduces a new two-variable p-adic L-function and proves an explicit reciprocity law connecting it to big Heegner points, refining previous results and confirming cases of the BSD conjecture.

Findings

Relation between classical Heegner cycles and big Heegner points' specializations

Vanishing of Selmer groups of CM elliptic curves twisted by Artin representations

Refinement of earlier work on Heegner points and cycles

Abstract

In this paper, we prove an "explicit reciprocity law" relating Howard's system of big Heegner points to a two-variable $p$-adic $L$-function (constructed here) interpolating the $p$-adic Rankin $L$-series of Bertolini-Darmon-Prasanna in Hida families. As applications, we obtain a direct relation between classical Heegner cycles and the higher weight specializations of big Heegner points, refining earlier work of the author, and prove the vanishing of Selmer groups of CM elliptic curves twisted by 2-dimensional Artin representations in cases predicted by the equivariant Birch and Swinnerton-Dyer conjecture.